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AN INTRODUCTION TO THE p-ADIC

ALEXA POMERANTZ

Abstract. This paper introduces the p-adic numbers with an emphasis on comparison to the real numbers. It is mostly self-contained, but some basic knowledge in theory, analysis, topology, and geometry is assumed. We begin by defining the p-adic and using this metric to construct Qp and Zp. We then move on to and and writing p-adic expansions. Lastly, we discuss topology and geometry in Qp.

Contents 1. Introduction 1 2. The p-adic Metric 2 3. Construction of Qp and Zp 5 4. Sequences and Series in Qp 6 5. p-adic Expansions 8 6. The Topology on Qp 10 7. Geometry in Qp 12 Acknowledgments 14 References 14

1. Introduction The p-adic numbers, where p is any , come from an alternate way of defining the distance between two rational numbers. The standard distance , the Euclidean , gives rise to the real numbers. While the real numbers are more natural to most of us, this paper aims to present the p-adic numbers on an equal footing. For example, both fields are complete metric spaces. Unlike the real numbers, the p-adic numbers are an ultrametric space, leading to a number of fascinating but often counterintuitive results. The p-adic numbers are useful because they provide another toolset for solving problems, one which is sometimes easier to work with than the real numbers. They have applications in , analysis, , and more. One example is Hensel’s lemma for finding roots of a . Another is Mahler’s Theorem, a p-adic analog of the Stone-Weierstrass Theorem. Yet another is Monsky’s Theorem, a theorem about triangulating squares whose proof makes use of 2-adic numbers. The reader is encouraged to read more about these in their own time, but the rest of this paper will be focused on introducing p-adic numbers more broadly. We start off by introducing the p-adic metric and then employing it to construct the p-adic numbers using Cauchy sequences. We then move on to some unique results about sequences and series that do not appear in real analysis. This leads 1 2 ALEXA POMERANTZ into a discussion of p-adic expansions, which will help us think about p-adic numbers in a more concrete way. Next, we discuss the topology on Qp to get a sense for p-adic spaces. Finally, we look at how p-adic geometry differs from .

2. The p-adic Metric We begin by defining a general notion of absolute value for an arbitrary field K.

Definition 2.1. An absolute value on K is a function |·|: K → R≥0 that satisfies the following properties for all x, y ∈ K: (i) |x| = 0 if and only if x = 0, (ii) |xy| = |x||y|, and (iii) |x + y| ≤ |x| + |y| (Triangle Inequality). Moreover, an absolute value is non-Archimedean if it also satisfies the following property: (iv) |x + y| ≤ max{|x|, |y|} (Strong Triangle Inequality). An absolute value that does not satisfy property (iv) is Archimedean. Remark 2.2. We note that any function satisfying Property (iv) necessarily satisfies Property (iii).

Definition 2.3. A metric on K is defined by a distance function d: K × K → R≥0. An absolute value induces a metric defined by d(x, y) = |x − y| for all x, y ∈ K. A on which a metric is defined is called a . A set with a metric induced by a non-Archimedean absolute value is called an ultrametric space. Lemma 2.4. Let K be an ultrametric space, and let x, y ∈ K. If |x|= 6 |y|, |x+y| = max{|x|, |y|}. Proof. Without loss of generality, we assume that |x| > |y|. Since K is an ultra- metric space, |x + y| ≤ max{|x|, |y|} = |x| (Definition 2.1 and Definition 2.3). Also, x = (x + y) + (−y), so |x| ≤ max{|x + y|, |y|}. Since |x| > |y|, |x + y| > |y|. Thus, we have |x + y| ≤ |x| ≤ |x + y|. It follows that |x + y| = |x| = max{|x|, |y|}.  Exercise 2.5. Check that in a field with a metric induced by an absolute value, the following properties hold for all x, y, z ∈ K: • d(x, y) > 0 ⇐⇒ x 6= y, • d(x, y) = d(y, x), and • d(x, z) ≤ d(x, y) + d(y, z). We now begin to define the p-adic metric specifically. For the rest of the paper, let p be a fixed prime number.

Definition 2.6. The p-adic on Q is defined by a function vp : Q → Z ∪ {∞}. Let x ∈ Q where x 6= 0. If x ∈ Z, let vp(x) be the unique positive satisfying v (x) 0 0 x = p p x , where p - x . a For all nonzero x ∈ , we may write x = where a, b ∈ . Then we define Q b Z vp(x) = vp(a) − vp(b).

Lastly, we define vp(0) = +∞. AN INTRODUCTION TO THE p-ADIC NUMBERS 3

Remark 2.7. We can think of the p-adic valuation as describing how divisible by p a number is. In this way, it makes sense to define vp(0) = +∞ since 0 can be divided by p infinitely many times, yielding an integer (namely 0) each time. Exercise 2.8. Show that the p-adic valuation does not depend on the representa- a a0 0 0 tion of a (if b = b0 , then vp(a) − vp(b) = vp(a ) − vp(b )). a0 Exercise 2.9. Prove that if x ∈ , x = pvp(x) , where p a0b0. Q b0 - Lemma 2.10. The following properties hold for all x, y ∈ Q: (i) vp(xy) = vp(x) + vp(y) and (ii) vp(x + y) ≥ min{vp(x), vp(y)}. a c Proof. Let x, y ∈ . Then x = and y = , where a, b, c, d ∈ . Q b d Z

First, if x = 0 or y = 0, then vp(xy) = vp(x) + vp(y) = +∞. We therefore consider the case that both x and y are nonzero. Then

vp(x) = vp(a) − vp(b), and vp(y) = vp(c) − vp(d) (Definition 2.6). ac Moreover, since xy = , bd

vp(xy) = vp(ac) − vp(bd). Since a, c ∈ Z, v (a) 0 v (c) 0 0 0 a = p p a and c = p p c , where p - a and p - c (Definition 2.6). Then ac = pvp(a)a0 · pvp(c)c0 = pvp(a)+vp(c)a0c0. 0 0 0 0 Since p - a and p - c , p - a c . Thus, by Definition 2.6, vp(ac) = vp(a) + vp(c). By a similar argument, vp(bd) = vp(b) + vp(d). It follows that

vp(xy) = vp(ac) − vp(bd) = vp(a) + vp(c) − vp(b) − vp(d) = vp(x) + vp(y). Hence, the p-adic valuation satisfies the first property.

Next, if x + y = 0, then vp(x + y) = +∞ ≥ min{vp(x), vp(y)} trivially. We therefore consider the case that x + y 6= 0. Without loss of generality, we assume that vp(x) ≤ vp(y). Then vp(x) is finite (if this were not the case, then x + y = 0 since x = y = 0). By Definition 2.6,

vp(x) = vp(a) − vp(b).

Also, if y = 0, vp(x + y) = vp(x) = min{vp(x), vp(y)}. Thus, we assume that y 6= 0, so vp(y) = vp(c) − vp(d).

Since vp(x) ≤ vp(y), vp(a) − vp(b) ≤ vp(c) − vp(d). It follows that vp(a) + vp(d) ≤ vp(b) + vp(c). Equivalently, vp(ad) ≤ vp(bc) (by part (i)). Since ad ∈ Z and bc ∈ Z, v (ad) v (bc) ad = p p m and bc = p p n, where p - m and p - n (Definition 2.6). 4 ALEXA POMERANTZ

Then ad + bc = pvp(ad)m + pvp(bc)n.

vp(ad) vp(bc) vp(ad) Since vp(ad) ≤ vp(bc), p | p . It follows that p | ad + bc. Thus, vp(ad + bc) ≥ vp(ad). We now manipulate this inequality as follows, applying the result from part (i) as necessary:

vp(ad + bc) ≥ vp(ad)

=⇒ vp(ad + bc) ≥ vp(a) + vp(d)

=⇒ vp(ad + bc) − vp(d) ≥ vp(a)

=⇒ vp(ad + bc) − vp(b) − vp(d) ≥ vp(a) − vp(b)

=⇒ vp(ad + bc) − vp(bd) ≥ vp(a) − vp(b) ad + bc We recall that x + y = , so bd

vp(x + y) = vp(ad + bc) − vp(bd).

Thus, we have vp(x + y) ≥ vp(x) = min{vp(x), vp(y)}. In other words, the p-adic valuation satisfies the second property. 

Definition 2.11. Let the p-adic absolute value function |·|p : Q → R≥0 be defined by ( p−vp(x) if x 6= 0, |x|p = 0 if x = 0.

The p-adic absolute value induces the p-adic metric, denoted by dp. Remark 2.12. Since the p-adic valuation for nonzero rational numbers is always an integer, the p-adic absolute value takes a discrete set of values. Proposition 2.13. The p-adic absolute value is a non-Archimedean absolute value on Q.

Proof. Let x, y ∈ Q. First, we suppose that x = 0. Then |x|p = 0 by Definition −vp(x) 2.11. Now, we suppose that x 6= 0. Then |x|p = p by Definition 2.11. Since p 6= 0, |x|p 6= 0. Thus, the p-adic absolute value satisfies Definition 2.1(i).

Next, we check the second property. In the case that x = 0 or y = 0, |xy|p = |x|p|y|p trivially. Hence, we consider the case that x 6= 0 and y 6= 0. Then by Definition −vp(xy) −vp(x) −vp(y) −vp(x)−vp(y) 2.11, |xy|p = p , and |x|p|y|p = p p = p . By Lemma 2.10(i), −vp(xy) = −vp(x) − vp(y). Therefore, |xy|p = |x|p|y|p, and the p-adic absolute value satisfies Definition 2.1(ii).

Lastly, we check the third and fourth properties. in the case that x+y = 0, |x+y|p = 0 by definition. Then |x + y|p ≤ max{|x|p, |y|p} trivially. We consider the case that x + y 6= 0. Without loss of generality, we assume that |x|p ≥ |y|p. Then |x|p 6= 0, −vp(x+y) so x 6= 0. It follows that vp(x) ≤ vp(y). By Definition 2.11, |x + y|p = p −vp(x) and |x|p = p . By Lemma 2.10(ii), vp(x + y) ≥ min{vp(x), vp(y)} = vp(x). It −vp(x+y) −vp(x) follows that p ≤ p . Equivalently, |x + y|p ≤ |x|p = max{|x|p, |y|p}. Thus, the p-adic absolute value satisfies Definition 2.1(iv) and consequently Defi- nition 2.1(iii).  AN INTRODUCTION TO THE p-ADIC NUMBERS 5

75 16 Exercise 2.14. Find , , and d11(89, 4082). 73 5 81 3 Exercise 2.15. Prove that lim pn = 0. n→∞ Proof. Let  > 0. By the Archimedean Property, there exists N ∈ N such that 1 1 1 1 1 < . Since pN > N, < < . Let n ≥ N. Then ≤ < . We note N pN N pn pN n n n −n n that p 6= 0 and vp(p ) = n, so |p |p = p . Hence, we have |p |p < . It follows that lim pn = 0. n→∞  Loosely stated, two absolute values are equivalent if they induce the same topol- ogy. Before moving on to the construction of the p-adic numbers, we state the following result about absolute values on Q. Theorem 2.16. (Ostrowski’s Theorem) Every nontrivial absolute value on Q is equivalent to either the standard absolute value or one of the p-adic absolute values. Proof. See [3, p. 56-59].  Because of Ostrowski’s Theorem, the p-adic absolute value can be viewed as just as important as the standard absolute value. Together, they comprise all possible absolute values on Q. There are a number of results that we will prove later that could apply to any ultrametric space, but we care about the p-adic absolute values in particular because they are essentially the only non-Archimedean absolute values on Q.

3. Construction of Qp and Zp Next, in to construct the field of p-adic numbers from the rational numbers, we use a process similar to the construction of the real numbers using Cauchy sequences. We will not include all of the details but aim to provide an outline for this process by including the most relevant definitions and results.

Definition 3.1. A (an) is Cauchy if for every  > 0, there is some N ∈ N such that for all m, n ≥ N, |an − am| < . Definition 3.2. A metric space X is complete under a given metric if every in X converges to a point in X. Proposition 3.3. The field of rational numbers, Q, is not complete under the p-adic metric. Proof. See [3, p. 63-64].  Theorem 3.4. Let K be a field with an absolute value |·|. Then there exists a complete field K0 with an absolute value |·|0 that extends K. This completion K0 is unique up to . Moreover, on K, |·|0 restricts to |·|. Lastly, K is dense in K0. Sketch of Proof. We omit this proof since it involves some abstract algebra that is outside the scope of this paper. The reader can see [9, p. 5-6]. The general idea is for each element of K0 to be represented by the limit of a Cauchy sequence of elements in K (or multiple equivalent Cauchy sequences). Any Cauchy sequence 0 0 that does not have a limit in K will have a limit in K , making K complete.  6 ALEXA POMERANTZ

Definition 3.5. Let the field of p-adic numbers, Qp, be defined by the completion of Q with respect to the p-adic metric. We know that such a completion exists and is unique (up to isomorphism) due to Theorem 3.4. Remark 3.6. We note that due to Ostrowski’s Theorem, the real numbers and the p-adic numbers are the only completions of the rational numbers.

We have now constructed Qp analytically and can construct Zp from it. Before defining Zp, we note that an algebraic construction also exists, in which Zp is constructed first and then Qp is constructed as its field of . We leave this alternate construction for the reader to explore.

Definition 3.7. Let the of p-adic , Zp, be defined as follows:

Zp = {x ∈ Qp | |x|p ≤ 1}.

Remark 3.8. We note that for all x ∈ Z, vp(x) ≥ 0, so |x|p ≤ 1. Thus, Z ⊂ Zp (just as Q ⊂ Qp).

Before moving on, we note, without proof, that Zp is also the completion of Z with respect to the p-adic metric (see [10]). Under the standard metric, Z is already complete, as the only Cauchy sequences in Z are constant sequences. This is because every integer is a distance of at least one away from another integer. However, under the p-adic metric, integers can get arbitrarily close to each other, so there are nontrivial Cauchy sequences. We will see an example of such a sequence in Section 4 (Example 4.4).

4. Sequences and Series in Qp

In this section, we will take a further look at sequences and series in Qp. We will see that they are often easier to work with than sequences and series in R. Because Qp is an ultrametric space, we can obtain some nice results that only apply partially in R.

Theorem 4.1. Let (an) be a sequence in Qp. Then (an) is Cauchy if and only if for every  > 0, there is some N ∈ N such that for all n ≥ N, |an+1 − an|p < .

Proof. First, we assume that (an) is Cauchy. Let  > 0. By Definition 3.1, there is some N ∈ N such that for all m, n ≥ N, |an − am|p < . Let n ≥ N. Then n + 1 ≥ N, so |an+1 − an|p < .

Next, we assume that for every  > 0, there is some N ∈ N such that for all n ≥ N, |an+1 − an|p < . Let  > 0. Then there is some N ∈ N such that

for all n ≥ N, |an+1 − an|p < .

Let m, n ≥ N. If n = m, |an − am|p = |0|p = 0 < , so we consider the case that m 6= n. Without loss of generality, we assume that n > m. Then n = m + k for some k ∈ N. We can rewrite |an − am|p as

|(am+k − am+k−1) + (am+k−1 − am+k−2) + ... + (am+1 − am)|p. Since the p-adic absolute value is non-Archimedean,

|an − am|p ≤ max{|am+i − am+i−1|p | 1 ≤ i ≤ k} (Definition 2.1(iv)). AN INTRODUCTION TO THE p-ADIC NUMBERS 7

For some 1 ≤ i0 ≤ k,

|am+i0 − am+i0−1|p = max{|am+i − am+i−1|p | 1 ≤ i ≤ k}.

Then |an−am|p ≤ |am+i0 −am+i0−1|p. We recall that m ≥ N, so by our assumption,

|am+i0 − am+i0−1|p < .

Hence, |an−am|p <  by transitivity. Therefore, (an) is Cauchy (Definition 3.1).  ∞ X Corollary 4.2. Let (an) be a sequence in Qp. Then an converges if and only n=0 if lim an = 0. n→∞

Proof. Let (pn) be the sequence of partial sums for (an).

∞ P First, we assume that an converges. Then (pn) converges. It follows that n=0 (pn) is Cauchy. Let  > 0. By Theorem 4.1, there is some N ∈ N such that for all 0 n ≥ N, |pn+1 − pn|p < . Equivalently, |an+1|p <  for all n ≥ N. Let N = N + 1. 0 Then for all n ≥ N , |an|p < .

Next, we assume that lim an = 0. Then there is some N ∈ such that for n→∞ N all n ≥ N, |an|p < . Then also, |an+1|p <  for all n ≥ N. Equivalently, |pn+1 − pn|p <  for all n ≥ N. By Theorem 4.1, (pn) is Cauchy and therefore ∞ P converges. Hence, an converges.  n=0 Remark 4.3. We recall that Theorem 4.1 and Corollary 4.2 are only true in the forward direction in R. In particular, the harmonic series is a well-known coun- terexample for the other direction. The harmonic sequence’s terms get arbitrarily close to 0, but the corresponding series diverges. ∞ X 1 Example 4.4. The p-adic series pn converges, and its sum is . 1 − p n=0 ∞ P n Proof. Let (an) be the sequence of partial sums for p . Then for all n ∈ N∪{0}, n=0 n−1 n−1 P i n n P i an = p . We recall the algebraic fact that 1 −p = (1−p) p . Equivalently, i=0 i=0 n−1 n P i 1 − p 1 1 n an = p = . We note that lim = and lim p = 0 (Exercise i=0 1 − p n→∞ 1 − p 1 − p n→∞ 2.15). By the algebra of limits,

1 n 1 1 1 lim an = lim − lim p · lim = − 0 = . n→∞ n→∞ 1 − p n→∞ n→∞ 1 − p 1 − p 1 − p ∞ P n 1 Therefore, p = .  n=0 1 − p Remark 4.5. This result is analogous to the formula for the sum of a geometric series in R. The result in R holds for all x ∈ R where |x| < 1. Similarly, we could extend this result to hold for all x ∈ Qp where |x|p < 1, rather than only for x = p. We leave this as an exercise for the reader. 8 ALEXA POMERANTZ

5. p-adic Expansions Using the formula in Example 4.4, we can obtain some fairly counterintuitive results. For example, in Q2, −1 = 1 + 2 + 4 + ··· . We aim to make sense of these results and think about p-adic numbers more concretely by introducing the idea of a p-adic expansion. First, we state the following proposition.

∞ X n Proposition 5.1. Any series in the form anp , with an ∈ {0, 1, ··· p − 1} and

n=n0 n0 ∈ Z, converges in Qp. n Proof. We recall from Example 2.15 that lim p = 0. We suppose that 0 ≤ an < p n→∞ ∞ n P n for all n ≥ n0. It follows that lim anp = 0 as well. By Corollary 4.2, anp n→∞ n=n0 converges in Qp (the starting index being n0 rather than 0 does not make a difference to whether the series converges). 

Now that we know that any series in this form converges p-adically, we can define the p-adic expansion.

Definition 5.2. The p-adic expansion of a number α ∈ Qp is a series in the form ∞ X n α = anp ,

n=n0

where n0 ∈ Z ∪ {∞}, an0 6= 0, and 0 ≤ an < p for all n ≥ n0.

We call the integers an the coefficients of the expansion.

Remark 5.3. When α ∈ N, the p-adic expansion of α is the same as the base p representation of α (where there exists some N ∈ N such that the coefficients an = 0 for all n ≥ N). Proposition 5.4. Every p-adic number has a unique p-adic expansion.

Proof. See [3, p. 82-83]. 

∞ P n Exercise 5.5. Let α ∈ Q, and let anp be the p-adic expansion of α. Show n=n0 that vp(α) = n0 (we note that for elements of Qp \Q, the p-adic valuation is defined this way).

Remark 5.6. We can write p-adic expansions in the same way that we write standard representations of numbers. However, in a p-adic expansion, there are a finite number of negative powers of p, as opposed to a finite number of positive powers of 10. Thus, if we write a p-adic expansion in the usual way, where the powers of p decrease from left to right, the coefficients often extend infinitely to the left.

Examples 5.7. Written in the form described in Remark 5.6, −1 = 12, 175 = 194 1200 , and = 36.5 (subscripts are used to specify the prime). 5 7 7 AN INTRODUCTION TO THE p-ADIC NUMBERS 9

Before moving on, we go back to our statement that in Q2, −1 = 1 + 2 + 4 + ··· . We can make sense of this by adding 1 as follows:

1 + (−1) = 1 + 20 + 21 + 22 + ··· ⇐⇒ 0 = 0 · 20 + 21 + 21 + 22 + ··· = 0 · 20 + 0 · 21 + 22 + 22 + ··· = 0 · 20 + 0 · 21 + 0 · 22 + ··· .

If we continue, every coefficient in the expansion would eventually become 0, mak- ing it easier to believe that this series does in fact represent the of 1.

Next, in order to help us compute p-adic expansions, we introduce the following lemma.

∞ X n Lemma 5.8. Let α ∈ Zp with p-adic expansion anp , and let k ∈ N. Then n=n0 k−1 P n k α ≡ anp (mod p ). n=n0

∞ ∞ P n k P n k Proof. We begin with α ≡ anp (mod p ). Also, anp ≡ 0 (mod p ) since n=n0 n=k k−1 k m P n k for all m ≥ k, p | p . Lastly, we subtract to obtain α ≡ anp (mod p ).  n=n0

4 Example 5.9. Determine the 5-adic expansion of . 3

  ∞ 4 4 P n Proof. Since vp = 0, = an5 , where 0 ≤ an ≤ 4 (Definition 5.2 and 3 3 n=0

4 4 4 Exercise 5.5). Also, since = 1, is a 5-adic integer. By Lemma 5.8, ≡ a0 3 p 3 3 (mod 5). Then 4 ≡ 3a0 (mod 5). We multiply by 2 to obtain a0 ≡ 3 (mod 5). 4 Since 0 ≤ a ≤ 4, a = 3 as well. Next, by Lemma 5.8, ≡ a + 5a (mod 25). n 0 3 0 1 We solve this congruence as follows:

4 ≡ 3a0 + 15a1 (mod 25)

=⇒ 4 ≡ 3 · 3 + 15a1 (mod 25)

=⇒ 20 ≡ 15a1 (mod 25)

=⇒ 3a1 ≡ 4 (mod 5).

=⇒ a1 ≡ 3 (mod 5).

4 As before, a = 3 as well. Using similar methods, we can solve ≡ a + 5a + 25a 1 3 0 1 2 (mod 125) to obtain a2 = 1. After this, the pattern repeats, alternating between 3 4 and 1. Written in way described by Remark 5.6, the 5-adic expansion of is 133 . 3 5 10 ALEXA POMERANTZ

We can check this by multiplying by 3 as follows: 4 = 3 · 50 + 3 · 51 + 1 · 52 + ··· 3 =⇒ 4 = 9 · 50 + 9 · 51 + 3 · 52 + ··· = 4 · 50 + 51 + 4 · 51 + 52 + 3 · 52 + ··· = 4 · 50 + 0 · 51 + 52 + 52 + 3 · 52 + ··· = 4 · 50 + 0 · 51 + 0 · 52 + ··· . 4 Every non-zero power of 5 eventually vanishes, leaving only 4. Thus, · 3 = 4, as 3 we would expect.  We saw in the previous example that a rational number had an eventually pe- riodic p-adic expansion. It turns out that this is true in general, just as it is with decimal expansions in R. Theorem 5.10. A p-adic number has an eventually periodic p-adic expansion if and only if it is rational. Proof. See [1, p. 3-5].  We can also define and for p-adic numbers using p-adic expansions. These definitions are similar to how we add and multiply the decimal representations of real numbers, as we add the coefficients and carry any remainders. The reader can see [7, p. 1-3] for more rigorous definitions of these operations. These definitions can also be used to verify that Qp is a field and that Zp is a .

6. The Topology on Qp In this section, we discuss the topology induced by the p-adic metric in an attempt to better visualize the p-adic numbers. While the real numbers can be visualized as a line, it is not as simple with the p-adics. In particular, there is an ordering on R but not on Qp. Moreover, R is connected, whereas Qp is totally disconnected (this will be further explained and proven later). Before formalizing these details, we examine the image below as one way to visualize the 3-adic integers.

Figure 1. A visualization of the 3-adic integers ([6]) AN INTRODUCTION TO THE p-ADIC NUMBERS 11

Each contains three main sub-. The integers within a subcircle of a given circle have a particular 3-adic distance from integers in another subcircle of that circle. For example, the orange circle has purple subcircles. The integers within a purple subcircle are a distance of one away from the integers in a different purple subcircle. The distance corresponding to the largest circle is one since one is the largest possible 3-adic distance between integers. While there are no circles larger than the orange one for Z3, the circles can get infinitely small, as we can choose numbers that differ by larger and larger powers of 3. In this way, the p-adic integers are not discrete even though they are disconnected. Hence, a good image of the p-adic integers would be fractal-like, as Figure 1 is. We now proceed to introduce and prove several results about the topology in- duced by the p-adic metric.

Theorem 6.1. In Qp, any open ball is also a closed ball (and vice versa).

Proof. Let B(a, r) be an open ball in Qp. Then B(a, r) = {x ∈ Qp | |x − a|p < r}. Let n be the smallest integer such that r ≤ p−n. For the reason described in Remark 2.12, there are no p-adic numbers with absolute value between p−(n+1) and p−n. −n −(n+1) Thus, B(a, r) = {x ∈ Qp | |x − a|p < p } = {x ∈ Qp | |x − a|p ≤ p }. It follows that B is also a closed ball. A symmetric argument would show that any closed ball is also an open ball in Qp. 

Corollary 6.2. The ring Zp is both closed and open in Qp.

Proof. We recall from Definition 3.7 that Zp = {x ∈ Qp | |x|p ≤ 1}. Thus, it is clear that Zp is a closed ball and is therefore closed. By Theorem 6.1, Zp is also an open ball and is therefore open (specifically, Zp = {x ∈ Qp | |x|p < p}). It follows that Zp is closed and open in Qp. 

Corollary 6.3. The field Qp is totally disconnected.

Proof. Let a and b be distinct points in Qp. Let δ = |a − b|p. Let A = {x ∈ Qp | |x − a|p < δ}. Then A is an open ball in Qp, and a ∈ A. For the same reason that Zp is both closed and open, A is both closed and open. Therefore, B := Qp \ A is also both closed and open. Since |b − a|p = δ, b∈ / A. Thus, b ∈ B. Since Qp = A ∪ B, where A and B are disjoint open sets with a ∈ A and b ∈ B, Qp is totally disconnected. 

Theorem 6.4. The ring Zp is compact. Proof. We recall that a metric space is compact if and only if it is sequentially com- pact. Thus, we aim to show that Zp is sequentially compact using a diagonalization argument.

∞ ∞ P i Let (αn)n=0 be a sequence in Zp. For each n ∈ N ∪ {0}, αn = an,ip , where i=0 0 ≤ an,i < p. This is clear from Definition 5.2. For elements of Zp, the starting index of the p-adic expansion is greater than or equal 0, but we can obtain an equivalent sum starting at index 0 by adding 0-terms to the beginning. By the Pigeonhole Principle, there exists b0 ∈ N ∪ {0} such that 0 ≤ b0 ≤ p − 1 and b0 = an,0 for infinitely many n. We construct a subsequence (α0n) of (αn), where (α0n) consists of the elements of (αn) for which an,0 = b0. We now inductively construct a sequence of subsequences. For each k ∈ N, we will construct a sequence 12 ALEXA POMERANTZ

(αkn) that is a subsequence of (α(k−1)n). Let k ∈ N. Then as before, there exists bk ∈ N ∪ {0} such that 0 ≤ bk ≤ p − 1 and bk = an,k for infinitely many n where ∞ P i αn = an,ip ∈ (α(k−1)n). We note that we have added an additional assumption i=0 so that we are not selecting elements of (αn) that were not present in (α(k−1)n). Now, let (αkn) be the subsequence of (α(k−1)n) that consists of every element for ∞ P i which an,k = bk. Let β = bip (so β ∈ Zp). Then for each k ∈ N ∪ {0}, the i=0 first k + 1 coefficients in the p-adic expansion of every element of (αkn) match the first k + 1 coefficients in the p-adic expansion of β. We finally consider the diagonal subsequence (αkk). This sequence converges to β by construction. Since (αkk) is a convergent subsequence of (αn), any sequence in Zp must contain a convergent subsequence. Hence, Zp is sequentially compact and is therefore compact. 

Corollary 6.5. The field Qp is locally compact.

Proof. Let x ∈ Qp. Let X = {x + y | y ∈ Zp}. Since Zp is compact (by Theorem 6.4), X is compact as well. Just as Zp is a ball centered at 0, X is a ball centered at x. Thus, X is a compact neighborhood of x, and Qp is locally compact. 

7. Geometry in Qp

We close by proving several interesting geometric results that hold in Qp.

Theorem 7.1. In Qp, all triangles are isosceles.

Proof. Let a, b, and c be distinct points in Qp. Then dp(a, b), dp(b, c), and dp(a, c) are the lengths of the sides of the triangle determined by those points. If dp(a, b) = dp(b, c), the triangle is isosceles. If dp(a, b) 6= dp(b, c), |a − b|p 6= |b − c|p. We note that (a − b) + (b − c) = a − c. By Lemma 2.4, |a − c|p = max{|a − b|p, |b − c|p}. In other words, either dp(a, c) = dp(a, b) or dp(a, c) = dp(b, c). Hence, the triangle is isosceles. 

Remark 7.2. We note also that if a triangle in Qp is not equilateral, its shortest side is the base of the triangle. This comes from the fact that the maximum function was used to find the two congruent sides. We note that we did not specify that the points of our triangle were not collinear. We see now that this cannot occur.

Corollary 7.3. No three distinct points in Qp are collinear. Proof. We assume, for the sake of contradiction, that there are distinct points a, b, and c that are collinear. Without loss of generality, we assume that dp(a, c) = dp(a, b) + dp(b, c). Then dp(a, c) > dp(a, b) and dp(a, c) > d(b, c) (since dp(a, b) > 0 and dp(b, c) > 0). Since ac is the longest segment, it follows from Theorem 7.1 and Remark 7.2 that ac is a leg of the isosceles triangle formed by the three points. However, that means dp(a, c) = dp(a, b) or dp(a, c) = dp(b, c). In either case, we have reached a contradiction. Thus, there are no three distinct points in Qp that are collinear.  AN INTRODUCTION TO THE p-ADIC NUMBERS 13

We now introduce another corollary.

Corollary 7.4. There are no right triangles in Qp. Proof. We assume, for the sake of contradiction, that there exist points a, b, c ∈ Qp such that 4abc is a right triangle. In other words, the side lengths of the triangle satisfy the . Without loss of generality, we assume 2 2 2 dp(a, c) = dp(a, b) + dp(b, c) . Then dp(a, c) > dp(a, b) and dp(a, c) > dp(b, c). Since every triangle in Qp is isosceles by Theorem 7.1, dp(a, b) = dp(b, c). Hence, we have a triangle in which the base is the longest side. This contradicts Remark 7.2. Therefore, there are no right triangles in Qp.  Lastly, we introduce a final theorem.

Theorem 7.5. At most p distinct points in Qp are equidistant from each other. Proof. We assume, for the sake of contradiction, that there exist at least p + 1 distinct points in Qp that are all equidistant from each other: a1, a2, . . . , ap+1. Let i, j ∈ N be such that 1 ≤ i, j ≤ p + 1 and i 6= j. Since the points are distinct, dp(ai, aj) 6= 0. It follows from Definition 2.1(i) and Definition 2.11 that −vp(ai−aj ) |ai − aj|p = p . Let vp(ai − aj) = m. Since ai, aj ∈ Qp, ai and aj have ∞ P n p-adic expansions. By Definition 5.2, ai = ai,np , where n0i ∈ Z, ai,n0i 6= 0, n=n0i ∞ P n and 0 ≤ ai,n < p for all n. Similarly, aj = aj,np , where n0j ∈ Z, aj,n0j 6= 0 n=n0j m and 0 ≤ aj,n < p for all n. We note that p | (ai − aj). It follows that for any k < m, ai,k = aj,k. Therefore, we can rewrite the difference as follows: ∞ ∞ X n X n ai − aj = ai,np − aj,np

n=n0i n=n0j ∞ ∞ X n X n = ai,np − aj,np n=m n=m ∞ ∞ X n+m X n+m = ai,n+mp − aj,n+mp n=0 n=0 ∞ ∞ m X n m X n = p ai,n+mp − p aj,n+mp . n=0 n=0 ∞ ∞ 0 P n 0 P n Now, let ai = ai,n+mp , and let aj = aj,n+mp . Then we have n=0 n=0 m 0 0 ai − aj = p (ai − aj).

Since vp(ai − aj) = m, m 0 0 vp(p (ai − aj)) = m. By Lemma 2.10(i), m 0 0 m 0 0 0 0 vp(p (ai − aj)) = vp(p ) + vp(ai − aj) = m + vp(ai − aj). 0 0 0 0 Therefore, vp(ai − aj) = 0. Since i and j were arbitrary, we can construct ai and aj 0 0 for all 1 ≤ i, j ≤ p + 1 where i 6= j. In each case, vp(ai − aj) = 0. However, since there are p + 1 points and only p possible values for the coefficients in the p-adic 14 ALEXA POMERANTZ

expansions, there exist distinct 1 ≤ i0, j0 ≤ p + 1 such that ai0,m = aj0,m (by the Pigeonhole Principle). Then p | (a0 − a0 ), so v (a0 − a0 ) 6= 0. Hence, we have i0 j0 p i0 j0 reached a contradiction, so our assumption must be false. There are no more than p points that are all equidistant from each other.  Remark 7.6. This theorem helps capture the notion that it is easier for points to be equidistant p-adically. We leave verification that p points can be equidistant from each other in Qp as an exercise to the reader. By contrast, in R, at most 3 points can all be equidistant from each other. Because the options for the distance between points are more limited in Qp, up to p points can be equidistant (in every case except Q2, this is more than 3). Acknowledgments It is my pleasure to thank my mentors, Iris Yunxuan Li and Cindy Tan. When I was overwhelmed by all the possible topics, they both helped me narrow down my interests and suggested sources for me to look at. Although I did not end up using most of the topics they showed me, I enjoyed being exposed to different topics by them. I’d also like to thank Iris in particular for helping me so much with writing this paper. Her feedback was truly invaluable. Thank you to Professor Rudenko for leading the Apprentice Program and making lectures and problem sets that introduced me to a lot of cool topics. Lastly, thank you to Dr. Peter May for all the time and effort he put into directing the REU, especially in these difficult circumstances. I really appreciate that I could still spend much of my summer doing math despite the unfortunate fact that it had to be remote.

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